
'''simple docstring'''
from __future__ import annotations

from fractions import Fraction







def 		A_(     A						:		int		,     A						:		Optional[int]):
							return (
							    num != den and num % 10 == den // 10 and (num // 10) / (den % 10) == num / den
							)







def 		A_(     A						:		str):
							UpperCamelCase   			=     []
							UpperCamelCase   			=     11
							UpperCamelCase   			=     int('1' + '0' * digit_len)
							for num in range(snake_case__		,     snake_case__):
														while den <= 99:
																					if (num != den) and (num % 10 == den // 10) and (den % 10 != 0):
																												if is_digit_cancelling(snake_case__		,     snake_case__):
																																			solutions.append(f'''{num}/{den}''')
																					den += 1
														num += 1
														UpperCamelCase   			=     10
							return solutions







def 		A_(     A						:		Tuple = 2):
							UpperCamelCase   			=     1.0
							for fraction in fraction_list(snake_case__):
														UpperCamelCase   			=     Fraction(snake_case__)
														result *= frac.denominator / frac.numerator
							return int(snake_case__)


if __name__ == "__main__":
						print(solution())










"""simple docstring"""




from __future__ import annotations

from decimal import Decimal

from numpy import array



def        a__					(							snake_case__							)		->   list[list[float]]:
			lowerCamelCase      =   Decimal

			# Check if the provided matrix has 2 rows and 2 columns
			# since this implementation only works for 2x2 matrices
			if len(snake_case__							) == 2 and len(matrix[0]							) == 2 and len(matrix[1]							) == 2:
						# Calculate the determinant of the matrix
						lowerCamelCase      =   float(
						    d(matrix[0][0]							) * d(matrix[1][1]							) - d(matrix[1][0]							) * d(matrix[0][1]							)							)
						if determinant == 0:
									raise ValueError("""This matrix has no inverse."""							)

						# Creates a copy of the matrix with swapped positions of the elements
						lowerCamelCase      =   [[0.0, 0.0], [0.0, 0.0]]
						lowerCamelCase ,  lowerCamelCase      =   matrix[1][1], matrix[0][0]
						lowerCamelCase ,  lowerCamelCase      =   -matrix[1][0], -matrix[0][1]

						# Calculate the inverse of the matrix
						return [
						    [(float(d(snake_case__							)							) / determinant) or 0.0 for n in row] for row in swapped_matrix
						]
			elif (
			    len(snake_case__							) == 3
			    and len(matrix[0]							) == 3
			    and len(matrix[1]							) == 3
			    and len(matrix[2]							) == 3
			):
						# Calculate the determinant of the matrix using Sarrus rule
						lowerCamelCase      =   float(
						    (
						        (d(matrix[0][0]							) * d(matrix[1][1]							) * d(matrix[2][2]							))
						        + (d(matrix[0][1]							) * d(matrix[1][2]							) * d(matrix[2][0]							))
						        + (d(matrix[0][2]							) * d(matrix[1][0]							) * d(matrix[2][1]							))
						    )
						    - (
						        (d(matrix[0][2]							) * d(matrix[1][1]							) * d(matrix[2][0]							))
						        + (d(matrix[0][1]							) * d(matrix[1][0]							) * d(matrix[2][2]							))
						        + (d(matrix[0][0]							) * d(matrix[1][2]							) * d(matrix[2][1]							))
						    )							)
						if determinant == 0:
									raise ValueError("""This matrix has no inverse."""							)

						# Creating cofactor matrix
						lowerCamelCase      =   [
						    [d(0.0							), d(0.0							), d(0.0							)],
						    [d(0.0							), d(0.0							), d(0.0							)],
						    [d(0.0							), d(0.0							), d(0.0							)],
						]
						lowerCamelCase      =   (d(matrix[1][1]							) * d(matrix[2][2]							)) - (
						    d(matrix[1][2]							) * d(matrix[2][1]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[1][0]							) * d(matrix[2][2]							)) - (d(matrix[1][2]							) * d(matrix[2][0]							))
						)
						lowerCamelCase      =   (d(matrix[1][0]							) * d(matrix[2][1]							)) - (
						    d(matrix[1][1]							) * d(matrix[2][0]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][1]							) * d(matrix[2][2]							)) - (d(matrix[0][2]							) * d(matrix[2][1]							))
						)
						lowerCamelCase      =   (d(matrix[0][0]							) * d(matrix[2][2]							)) - (
						    d(matrix[0][2]							) * d(matrix[2][0]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][0]							) * d(matrix[2][1]							)) - (d(matrix[0][1]							) * d(matrix[2][0]							))
						)
						lowerCamelCase      =   (d(matrix[0][1]							) * d(matrix[1][2]							)) - (
						    d(matrix[0][2]							) * d(matrix[1][1]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][0]							) * d(matrix[1][2]							)) - (d(matrix[0][2]							) * d(matrix[1][0]							))
						)
						lowerCamelCase      =   (d(matrix[0][0]							) * d(matrix[1][1]							)) - (
						    d(matrix[0][1]							) * d(matrix[1][0]							)
						)

						# Transpose the cofactor matrix (Adjoint matrix)
						lowerCamelCase      =   array(snake_case__							)
						for i in range(3							):
									for j in range(3							):
												lowerCamelCase      =   cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
						lowerCamelCase      =   array(snake_case__							)
						for i in range(3							):
									for j in range(3							):
												inverse_matrix[i][j] /= d(snake_case__							)

        # Calculate the inverse of the matrix
						return [[float(d(snake_case__							)							) or 0.0 for n in row] for row in inverse_matrix]
			raise ValueError("""Please provide a matrix of size 2x2 or 3x3."""							)



